Question 1112637
The sketch below (not to scale) illustrates the situation.
{{{drawing(300,450,-5,17,-1.5,31.5,
triangle(0,0,2.5,19.843,16,0),
triangle(-4.25,29.765,2.5,19.843,3.75,29.765),
locate(-0.6,0,A),locate(15.8,0,B),
locate(2.7,20.5,C),locate(3.5,30.9,D),
locate(-4.4,30.9,E)
)}}} The sketch is purposely not to scale, because triangle DEC is so small compared to triangle ABC, that making DEC to scale would make it almost invisible.
Triangles ABC and DEC must have been designed by the surveyor
so as to be similar triangles.
Their angles at C are congruent because they are vertical angles.
sides EC and DC were made to be proportional to BC  and AC respectively:
{{{BC/EC=600ft/"12 ft"=50}}} and
{{{AC/DC=500ft/"10 ft"=50}}} .
Because there is obe pair of congruent angles at C,
flanked by pairs of proportional sides,
triangles ABC and DEC are similar triangles "by SAS similarity".
DEC is a scale down version of ABC,
with all lengths in  ABC being {{{50}}} times greater.
So, {{{AB/ED=50}}}
{{{AB/"8 ft"=50}}}
{{{AB=50(8ft)}}}
{{{highlight(AB=400ft)}}} .